Phenomena that arise when a quantity that vanishes according to the dynamical rules of classical physics acquires a finite value when quantum rules are used. For example, the classical Poisson bracket for some entities may vanish; yet the corresponding quantum commutator may be nonzero—this is a commutator anomaly. Alternatively, the flow of some material current may satisfy a continuity equation by virtue of the classical equations of motion, indicating conservative flow; but upon quantization the continuity equation may fail and the flow may no longer be conservative in the quantum theory—this is an anomalous divergence (of the current in question). Since the forms of Poisson brackets and quantum commutators as well as the occurrence of continuity equations for currents are related to symmetries and conservation laws of the theory, quantum anomalies serve to break some symmetries and destroy some conservation laws of classical models. This violation of symmetry is not driven by explicit symmetry-breaking terms in the dynamical equations—rather the quantization procedure itself violates the classical symmetry. The mathematical reason for this phenomenon is that classical dynamics, involving a finite number of degrees of freedom, usually leads to a quantum theory on an infinite-dimensional vector space (Hilbert space), and this “infinity” gives rise to novel effects. See also: Canonical coordinates and transformations; Conservation laws (physics); Equation of continuity; Hilbert space; Nonrelativistic quantum theory; Symmetry laws (physics)

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